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Spin glasses and Stein’s method
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  • Open Access
  • Published: 20 August 2009

Spin glasses and Stein’s method

  • Sourav Chatterjee1 

Probability Theory and Related Fields volume 148, pages 567–600 (2010)Cite this article

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Abstract

We introduce some applications of Stein’s method in the high temperature analysis of spin glasses. Stein’s method allows the direct analysis of the Gibbs measure without having to eate a cavity. Another advantage is that it gives limit theorems with total variation error bounds, although the bounds can be suboptimal. A surprising byproduct of our analysis is a relatively transparent explanation of the Thouless–Anderson–Palmer system of equations. Along the way, we develop Stein’s method for mixtures of two Gaussian densities.

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Acknowledgments

The author thanks Michel Talagrand, Persi Diaconis and the associate editor for various helpful suggestions. The author is also grateful to the referee for a very careful reading of the proofs and a large number of useful comments.

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Department of Statistics, University of California at Berkeley, 367 Evans Hall #3860, Berkeley, CA, 94720-3860, USA

    Sourav Chatterjee

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  1. Sourav Chatterjee
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Corresponding author

Correspondence to Sourav Chatterjee.

Additional information

The author’s research was partially supported by NSF grant DMS-0707054 and a Sloan Research Fellowship.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Chatterjee, S. Spin glasses and Stein’s method. Probab. Theory Relat. Fields 148, 567–600 (2010). https://doi.org/10.1007/s00440-009-0240-8

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  • Received: 27 November 2007

  • Revised: 20 July 2009

  • Published: 20 August 2009

  • Issue Date: November 2010

  • DOI: https://doi.org/10.1007/s00440-009-0240-8

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Keywords

  • Spin glass
  • Sherrington–Kirkpatrick model
  • TAP equations
  • High temperature solution
  • Stein’s method

Mathematics Subject Classification (2000)

  • 60K35
  • 82B44
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